Numerical Simulation of Oil Dispersions in Vertical Pipe Flow

42 Journal of the Japan Petroleum Institute, 53, (1), 42-54 (2010)

[Regular Paper]

Numerical Simulation of Oil Dispersions in Vertical Pipe Flow

Mahdi Parvini, Bahram Dabir＊, and Seyed Abolfazl Mohtashami

Dept. of Chemical Engineering, Amirkabir University of Technology, Hafez St., P.O. Box 15875/4413 Tehran, IRAN

(Received October 27, 2008)

Immiscible liquid dispersions are widely used in chemical process, petroleum industries, polymerization, hetero-geneous chemical synthesis, etc.　In most of these chemical processes, the rate of inter-phase heat and mass transfer is known to strongly affect the overall performance and depends on the interfacial contact area between the phases.　This study at CFD simulations of immiscible liquid dispersion has been performed on a vertical pipe.　With specific reference to dispersed liquid-liquid flows, it was seen that the two-fluid approach (Eulerian-Eulerian approach) was extensively used for multiphase modeling, especially when detailed predictions are desirable over a range of holdups in exchange for a reasonable amount of computation power.　The results of CFD predictions for water as a continuous and kerosene as a dispersed phase have been compared with the experiments of Farrar and Bruun and Al-Deen and Bruun.　These simulations have also been done to understand the effect of various signif-icant forces in turbulent liquid dispersions (drag, lift, turbulent dispersion and added mass).　Several expressions for these forces were tested in order to choose the best combination.　Further, the problem has been simulated using two different turbulence models.　It has been found that lift force is more important than turbulent dispersion and added mass.　Inter-phase closure guidelines for liquid-liquid bubbly flows were developed based on simulation results that yielded the best agreement with experimental data.

Keywords Numerical simulation, Computational fluid dynamics, Immiscible dispersion, Vertical pipe flow, Interface force

1.　Introduction CFD to accurately predict the hydrodynamics in most single-phase flows cannot be readily translated to the

In chemical process industries, liquid-liquid disper- case of multiphase dispersions. sions are commonly encountered, for example, in sol- The presence of the dispersed phase invariably intro-vent extraction, emulsification, polymerization, hetero- duces additional complexities.　Nevertheless, if funda-geneous chemical synthesis, direct contact heat transfer, mental multiphase characteristics such as the dispersed homogenization process.　For optimal design and effi- phase holdup could be accurately predicted, it would be cient operation of equipments handling such disper- easier to estimate the contact area and the dynamics of sions, it is imperative that a qualitative and quantitative the dispersed phase (size distributions, breakup, coales-understanding of the hydrodynamics to be developed.　 cence, etc.). Also the mass and heat transfer between the dispersed There are several known approaches to model dis-and continuous phases assumes additional importance persed two-phase flows using CFD.　Some of the com-in the production, processing and transport of products.　 monly used ones are the two-fluid (Eulerian-Eulerian), Experiments can yield significant insight into the fac- discrete phase (Eulerian-Lagrangian), and interface tors affecting the interfacial area.　However, problems tracking (volume of fluid) approaches.　Amongst the such as limited range of applicability of the empirical aforementioned approaches, the two-fluid approach is correlations so developed, difficulty in extrapolating the widely used.　Although there are several studies in inferences to other geometries or even scaled-up versions context of turbulent bubbly flows, a large number of of the same equipment, and overall cost and time frame these studies are found on gas-liquid systems (Friberg1); limitations have led to more attention being focused on Thakre and Joshi2); Joshi3); Sokolichin et al.4); Deen et numerical approaches such as computational fluid al.5); Dhotre and Joshi6)).　It is well known that con-dynamics (CFD).　This is particularly true for multi-fluid cepts and results related to gas-liquid systems cannot be flow in complex geometries.　The proven success of readily applied to liquid-liquid systems owning to small

density differences and high viscosity ratios (Jin et al.7); ＊ To whom correspondence should be addressed. Brauner8); Shi et al.9); Madhavan, S.10)).　Because the ＊ E-mail: [email protected] closure requirements for immiscible liquid dispersion

J. Jpn. Petrol. Inst., Vol. 53, No. 1, 2010

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systems has received relatively lesser attention, the present paper, therefore, attempts to fill this void by means of an organized study which clearly distinguishes and quantifies the various significant forces acting on the dispersed phase in immiscible liquid dispersions. 1. 1.　Treatment of Inter-phase Forces in Dispersed

Multi-fluid Flow It is well known that in dispersed multiphase flows,

inter-phase forces play a significant role in determining the local phase distribution (Jakobsen et al.11); Friberg1); Deen5); Joshi3); Ranade12); Worner13); Chen et al.14); Tabib et al.15)).　Some of the typical interface forces encountered in multi-fluid dispersion are gravitational and buoyancy force, drag force, lift force, virtual/added mass force, Basset force, wall lubrication force and turbulent dispersion. 1. 2.　Liquid-liquid Dispersion in Vertical Flow

Most of the previous CFD related studies have focused on extraction columns (Rieger16); Modes and Bart17); Vikhansky and Kraft18)) or mechanically agitated tanks (Alopaeus et al.19); Agterof et al.20)).　These configura-tions suffer from the presence of complex hydro-dynamics with very large spatial variation in energy dis-sipation rates and feature a wide range of circulation times.　In this paper, the vertical pipe was chosen as the geometry of interest as it featured simple hydro-dynamics in addition to an extensive database of accu-rate experimental results (Farrar and Bruun21); Al-Deen and Bruun22)).

Liquid-liquid pipe flows are encountered in a broad range of operations such as the production, processing and transport of petroleum resources, direct contact heat transfer and various reaction engineering opera-tions.　Domgin et al.23) studied the turbulent dispersion of drops in a vertical pipe.　They used the Eulerian-Lagrangian approach to simulate the experimental work of Calabrese and Middleman24) on the turbulent disper-sion of drops in a vertical pipe (50.8 mm ID and 9.1 m long).　Hamad et al.25) simulated kerosene-water up-flows in a vertical pipe (78 mm ID and 1.5 m long) and compared their results with the experimental data of Farrar26). 1. 3.　Mult iphase and Turbulence Model ing

Approach Domgin et al.23) used the Eulerian-Lagrangian

approach in their work, as one of their objectives was to couple a lagrangian model to commercial CFD code and improve the basic k-ε formalism using algebraic stress relations for non-isotropic turbulent flow.　Their lagrangian model did not take into account dispersed fluid rotation, dispersed fluid collisions or turbulence modulation of the carrier (continuous) phase due to the presence of dispersed phase.　An alternative approach (called the Reynolds Stress Model or RSM) was also tested where the turbulence in the carrier flow was pre-dicted using second order turbulence model, which

would directly compute the fluctuating velocity correla-tions.　However, since the improvements brought by RSM turbulence model did not justify the additional CPU time requirements, it was not used in their final simulations.　Also, in their work the value of σk, a constant in the transport equation for the continuous phase turbulent kinetic energy was modified from its default value of 1 to 2.3, in order to obtain a numerical profile for the turbulent kinetic energy which was in good agreement with the experimental data of Sabot27)

for turbulent pipe flow.　In another investigation, Hamad et al.25) used the two-fluid Eulerian-Eulerian model to predict the dispersed phase volume fraction and phase velocity distribution for kerosene-water up-flows in a vertical pipe.　Their preliminary single phase predictions (velocity and turbulence intensity profiles) reported very good agreement with experimen-tal data.　In their liquid-liquid CFD simulations, the velocity profiles for both phases were predicted quite accurately.　However, they were unable to precisely predict the distribution of the dispersed phase volume fraction, particularly the characteristic near-wall peak.　They attributed this discrepancy to uncertainty in the experimental data as well as the insufficient treatment of inter-phase forces.　While transport equations for the turbulent kinetic energy and turbulent energy dissi-pation rate were solved for the continuous phase, there was no mention of how the turbulence in the dispersed phase was accommodated.　It was, however, mentioned that their k-ε model did not take into account drop-induced turbulence (e.g. similar to the models developed by Lance and Bertodano28) and Bertodano et al.29) for bubbles). 1. 4.　Treatment of Drop Diameter

Calabrese and Middleman24) had observed in their experiments that the drop diameter did not have a sig-nificant influence on the radial dispersion.　As a result, they proposed a single curve for the radial drop disper-sion irrespective of the drop diameter.　Domgin et al.23)

used a constant uniform drop diameter in all their simu-lations which were based on the experimental condi-tions of Calabrese and Middleman24).　Interestingly, it was seen from the simulations of Domgin et al.23) that unlike the experimental observations of Calabrese and Middleman24), the degrees of dispersion for different diameter drops were clearly distinct, particularly for drops more dense than water.　In the other study by Hamad et al.25) there was no mention of how the drop diameter was treated in their CFD simulations. 1. 5.　Interface Forces

Both Domgin et al.23) and Hamad et al.25) used the standard drag expression for a single rigid sphere throughout their simulation.　Hamad et al.25) considered non-drag inter-phase forces to be minor additional forces and consequently did not account for them in their sim-ulations.　On the other hand, Domgin et al.23) studied


44

the effect of virtual mass, pressure effects, and the bas-set (or history) terms in their simulations.　They justi-fied the inclusion of the virtual mass force in their study on the basis of a wide range of density ratios involved in their investigations.　Nevertheless, it was seen from their simulations that the basset and pressure terms had a weak influence on the results.　It should be noted that Domgin et al.23) also did not consider the effect of lift forces in their simulations.

2.　Mathematical Model

While there are several formulations of Eulerian-Eulerian approach, the form of governing equations as proposed by Ishii30) is most commonly used in fluid-fluid flows.　The basis for the governing equations has been discussed in detail by Van Wachem and Almstedt31).　The two-fluid Eulerian-Eulerian approach is used in the current paper solving the following conservation equa-tions: 2. 1.　Conservation of Mass (equation of continuity)

Conservation of mass for any phase (q) is given by the following transport equation:

∂ dρq (α ρq ) + ∇.(α ρqυq ) = m pq − αqq q ∑n

(1)∂t dt p=1

The solution of this equation for each phase, consider-ing the condition that the phase volume fractions should sum to unity, allows for the calculation of individual phase volume fractions. 2. 2.　Conservation of Momentum

The conservation of momentum for a continuous fluid phase ‘q’ in a non-accelerating frame of reference is given by:

∂ ∂t ( q q q q q q q q q qq qα ρ υ ) + ∇.(α ρ υ υ ) = −α ∇P + ∇.τ + α ρ g +

(Fdrag,q + FLift,q + FVM,q ) + ∑n

(Kpq (υp − υq ) ++ m pqυpq )(2)

p=1

Where the q-th phase stress tensor, tq, for laminar flow is given by:

2 α λ υυ υ∇ + ∇ −q q q q q q q q

T

∇ (3)τ = α µ ( ) + µ I 3

Here μq and λq are the shear and bulk viscosity of phase ‘q.’　The relative velocity between the continu-ous and dispersed phase is given by υ

− υ ), where ( q

‘ I ’ is a unit tensor and ‘p’ is the pressurep

shared by all phases.　The bulk viscosity is typically ignored in fluid-fluid flows as it assumes importance only in gas-solid flow and flow that feature shock waves, absorption and attenuation of acoustic waves, etc. (Rande12)).

For steady-state incompressible flow in the absence of mass transfer, external body forces such as the cen-trifugal forces, and virtual mass effects (which assume

significance only when high-frequency fluctuations of the relative velocity are predominant; Drew32); Chen et al.14)), the momentum conservation equation simplifies to:

∇.(α ρ υ υq ) = − q∇P + ∇τq q q α . q +

(4)α ρqg + F ++ ∑ K ( ))=p 1

The forces indicated above represent the inter-phase

q Lift,q ( −n

pq p qυ υ

drag force, lift force, virtual mass force and turbulent dispersion force, respectively.　In the present hydro-dynamic model all forces have been used.　A brief description of each interfacial force component is presented below.

The origin of the drag force is due to the resistance experienced by a body moving in the liquid.　Viscous stress creates skin drag and pressure distribution around the moving body creates form drag.　The latter mecha-nism is due to inertia and becomes significant as the dispersed fluid Reynolds number becomes larger.　The inter-phase momentum transfer between gas and liquid due to drag force is given by:

3 CD = − q υp − υq (υp − υq ) (5)Fdrag,q α ρq4 dp

Where, CD is the drag coefficient taking into account the character of the flow around the bubble and dp is the dispersed fluid diameter, the lift force arises from the net effect of pressure and stress acting on the surface of a bubble.　The lift force in terms of the slip velocity and the curl of the liquid phase velocity can be de-scribed as:

F = CLα ρq (υp − υqLift,q q )∇ × υq (6)

Where, CL is the lift coefficient; the sign of this force depends on the orientation of slip velocity with respect to the gravity vector.

When a bubble or drop moves in a liquid field with a non-uniform velocity, it accelerates some of the liquid in its neighborhood.　Due to the acceleration induced by the bubble or drop motion, the surrounding liquid experiences an extra force, which is due to the motion in a non-inertial frame of reference, called virtual or added mass force.　Then the virtual mass force is pro-portional to relative phase accelerations as follows:

dυq dυp F = −F = C α ρ −VM,q VM,p VM q q (7) dt dt

This force was neglected in a l l the previous simulations in accordance with the observations made by Hunt et al.33), Thakre and Joshi2), Deen et al.5), and Sokolichin et al.4).

The turbulent dispersion force, derived by Lopez de Bertodano34) is based on the analogy with molecular movement.　It approximates a turbulent diffusion of the bubbles by the liquid eddies.　It is formulated as:


45

FTD = −CTD ρq k∇αq (8)

Where, k is the liquid turbulent kinetic energy per unit of mass.　CTD is the turbulent dispersion coefficient. 2. 3.　Turbulence Equations

The Reynolds-averaged Navier-Stokes (RANS) equa-tions govern the transport of the averaged tow quanti-ties, with the whole range of the scales of turbulence being modeled.　Entire hierarchies of closure models are available in FLUENT including Spalart-Allmaras, k-ε and its variants, k-ω and its variants, and the RSM.

Large Eddy Simulation (LES) provides an alternative approach in which large eddies are explicitly computed (resolved) in a time-dependent simulation using the Navier-Stokes equations.　This model was not consid-ered in present paper. 2. 3. 1.　k-ε Turbulence Model

One of the most prominent turbulence models, the k-εmodel, has been implemented in most general purpose CFD codes and is considered the industry standard model.

In this model continuous phase turbulence is usually modeled using transport equations of ‘k’ and ‘ε’ which are written in a form similar to those found in single phase flow:

∂ µt,q (α ρ ϕ ) + ∇.(α ρ υ ϕ ) = ∇. α ∇ϕ ++ Sϕqq q q q q q q q q (9)∂t σϕ

Where, φq can be the turbulent kinetic energy or the tur-bulent energy dissipation rate of the continuous phase ‘q.’　The symbols μt,q and σφ refer to the turbulent vis-cosity and the turbulent Prandtl number for ‘k’ and ‘ε’ of the continuous phase.

The term ‘Sφq’ in Eq. (9) is the source term for the quantity, φ.　This can be further expanded as below:

S = α G − α ρ ε + α ρ Πkkq q kq q q q q q q (10)

S = α εkq

q [C C − C ρ ε ] + α ρ Πεεq q ε1 kq ε2 q q q q q (11)

Here, Ckq represent the turbulent kinetic energy gen-eration due to mean velocity gradients in phase ‘q,’ and is again calculated in a fashion similar to single-phase flows, where:

tU jGkq � ´ i´ (12)RU U j txi

Extra generation or dissipation of turbulence due to the presence of the dispersed phase (i.e. turbulence modulation) is accounted for in the last terms of expres-sions for Skq and Sεq respectively (i.e. αqρqΠkq and αqρqΠεq).　There have been several attempts to develop models in order to represent such extra terms and these were reviewed by Lahey35) and recently by Peirano and Leckner36).

The continuous phase turbulent viscosity is calculated

Table 1　Standard k-ε Turbulence Model Constants (Madhavan10))

Model parameter Default (suggested) value

Cε1 1.44Cε2 1.92Cm 0.09sk 1se 1.3

using an expression similar to that used in single phase flows:

kq2

µt,q = ρqC (13)µ εq

The standard k-ε turbulence model contains certain em-pirical constants which are listed in Table 1. 2. 3. 2.　R e y n o l d s S t r e s s M o d e l i n g ( R S M )

Turbulence Model In flows where the turbulent transport or non-

equilibrium effects are important, the eddy-viscosity assumption is no longer valid and results of eddy-viscosity models might be inaccurate.　In the RSM model, individual Reynolds stresses are computed via a differential transport equation.　The exact form of Reynolds stress transport equations is derived by taking moments of exact momentum equation.　Thus, the RSM model solves six Reynolds stress transport equa-tions.　Along with these, an equation for dissipation rate is also solved.　The exact transport equation for the transport of Reynolds stresses is given by:

t � ´ ´ � t �A R U U U´ � A P’ � A & �A R U U ´ �q q i j q q k i j q ij q ijtt txk2 (14)¤ ´ i´ j 2t ¤ 2 k ³ tU UA M � C ’R D A R Eq q s ´ ´

³ij q q qtxk ¦¥ ¦¥ 3 Eq µ txk µ 33

where Φij is the pressure-strain correlation, and P’, the exact production term, is given by:

P’ � R i j( )�U U U´ ´qT ( ) iU (15)� � �U U ´ j

As the turbulence dissipation appears in the individual stress equations, an equation for ε is computed with the model transport equation:

t t t ¤ ¤ Mt,q ³ tEq ³�A R E � �A R U E � A M �q q q q q i q q q ´ ´ �tt txi txi ¦¥ ¦¦¥ SE µ tx j µ2 (16)

¤ tUi ³ Eqq EqAqC R ´ ´ k C qE1 q U Ui ´ E2R Aq¦¥ txk µ k k

k is calculated from the solved values of normal stress using the Reynolds stress transport equation, as

U U jk � 1 �£ � ,´ i´ (17)2 i 1 2 3,

Turbulence in the dispersed phase may be physically understood as dispersed fluid velocity fluctuation caused by inter-dispersed fluid collisions and inter-


46

actions of the dispersed fluids with the turbulent contin-uous phase (Balzer et al.37); Simonin38); Enwald et al.39)).　However, if the concentration of dispersed phase is assumed to be dilute, inter-dispersed fluid collisions are negligible and the random motion of the dispersed phase is dominated by the turbulence in the continuous phase.　Fluctuating quantities of the dispersed phase can, therefore, be given in terms of the mean character-istics of the continuous phase and the ratio of the dis-persed fluid relaxation time and eddy-dispersed fluid interaction time.

A Reynolds Stress Model may be more appropriate for flows with sudden changes in strain rate or rotating flows. 2. 4.　 Effect of Turbulent Dispersion

Turbulent Dispersion Force results in additional dis-persion of phases from high volume fraction regions to low volume fraction regions due to turbulent fluctua-tions.　This is caused by the combined action of turbu-lent eddies and inter-phase drag.　For example, in a dispersed two-phase flow, dispersed particles get caught up in continuous phase turbulent eddy, and are trans-ported by the effect of inter-phase drag.　The effect is to move particles from areas of high to low concentra-tion.　Hence, this effect will usually be important in turbulent flows with significant inter-phase drag.

In the context of dispersed liquid-liquid systems, the importance of turbulent dispersion has been recognized by quite a few investigators (Domgin et al.23); Soleimani et al.40); Hamad et al.25)).　As all models work towards homogenizing the dispersed phase distribution depending on the intensity of turbulence, the effect of turbulent dispersion has been demonstrated using just two models (Lopez and Favre)41).

In this study two models for simulating inter-phase turbulent dispersion force were used: 2. 4. 1.　Favre Averaged Drag Model

It is a model based on the Favre average of the inter-phase drag force:

υtq ∇αp ∇αq FTD,p = FTD,q = −CTD Cpq − (18) σ tq αp αq

This model has been shown to have a wide range of universality.　σtc is the turbulent Schmidt number for continuous phase volume fraction. 2. 4. 2.　Lopez de Bertodano Model

The model of Lopez de Bertodano34) was one of the first models for the turbulent dispersion force:

FTD,p = −FTD,q = −CTD ρq k∇αq (19)

There is not a universally valid value of the non-dimensional Turbulent Dispersion Coefficient.　Values of 0.1-0.5 have been used successfully for bubbly flow with bubble diameters of order of a few millimeters.

Fig. 1　A Typical Radial Grid Layout for a Geometry; 58,880 Cells (60,701 nodes)

3.　Numerical Computation

3. 1.　Numerical Method To solve governing equations, the finite-volume

scheme based on the FLUENT42) code was employed and the SIMPLE method was applied for the pressure-velocity coupling.　Moreover, second-order central dif-ference was used for convective terms as inherent ability of second order schemes to suppress any non-physical ‘wiggles,’ in the CFD solution.　As liquid-liquid vertical pipe flows feature steady state behavior, therefore, steady state conditions were assumed.　A double-precision solver was employed in place of a single precision solver as disparate length scales were encoun-tered in simulations.　A three-dimensional model is used for all simulations.　The standard k-ε model was used to account for turbulence.

Velocity inlet boundary condition was applied to de-fine the flow velocity, along with all relevant scalar properties of the flow.　A constant, uniform velocity, normal to the inlet boundary was specified in all the simulations.　A pressure outlet boundary condition was used at the pipe exit.　In the present study, a zero gauge pressure was specified at the outlet.　Along the walls, no-slip boundary conditions were adopted.

A typical grid layout is shown in Fig. 1.　As the geometry being modeled (cylindrical pipe) was quite simple, a structured hexahedral grid was used.　The grid independence study was carried out using two grid resolutions.　The grid sizes used for the independence system in simulating experiments were 58,880 and 120,000. 3. 2.　Experimental Details

Farrar and Bruun21) carried out experiments in a 1.5 m long pipe of 78 mm internal diameter with kerosene-water system.　Hot film anemometry (HFA) was used for measuring the profile of holdup and velocity.　Similar work was done by Al-Deen and Bruun22) in a 1.5 m long pipe of 77.8 mm internal diameter (D).　The distance between the entrance and the axial testing position was 16D (1.25 m) of straight vertical pipe.　Summary of liquid-liquid data points and their experi-mental flow conditions are organized in Table 2.


47

Table 2　Summary of Experimental Flow Conditions for Selected Data Points

Pipe length and diameter Continuous Dispersed Average Total Equivalent

phase phase experimental volumetric drop

Data set Data point superficial superficial dispersed flow rate diameter L [m] ID [mm] velocity velocity phase

[mm] (Qp＋Qq)

[m/s] [m/s] holdup [—] [m3/s]

Farrar and F20 1.50 78 0.4935 0.1363 0.1912 5.00 0.00308

Bruun21)

Al-Deen and A05 1.50 77.8 0.5441 0.0286 0.0493 3.00 0.00272

Bruun22)

4.　Results and Discussion

For accurate prediction of local hydrodynamics, it is extremely important to properly select the simulation parameters of the interfacial forces like lift force, drag force, dispersion force and virtual mass force.　These selections should always be made based on the consid-erations of actual physics.　For example, a proper choice of drag law and dispersed fluid (bubble/droplet) diameter is needed to accurately predict the slip velocity.　So, it becomes important that one understands the inter-relation between drag forces, dispersed fluid size and slip velocity.　An attempt has been made here to eluci-date this point.

The slip velocity, which can be considered as the sig-nature of the multiphase system under a given condi-tion, is given as:

4dp ρq − ρp Vs = g (20) 3CD ρp

From Eq. (20), it is clearly seen that, for a given value of drag force, slip velocity changes with bubble/droplet size.　To understand this interrelation between drag forces, dispersed fluid size and slip velocity, slip velocity has been plotted as a function of dispersed fluid size for different drag laws (Fig. 2).

The list of drag laws considered and its expressions have been given in Table 3.　From Fig. 3, it can be seen that a single value of slip can be obtained from several combinations of drag law and dispersed fluid size.　Similarly, for a particular dispersed fluid size, one can obtain several values of slip depending on the drag law.　It is, therefore, of prime importance to choose the correct combination of drag law and dis-persed fluid size to model the gas-liquid flow in a dis-persed fluid column or the liquid-liquid flow in a verti-cal pipe.　Furthermore, with changes in superficial dispersed phase velocity, and the change in nature of two phase systems, the average dispersed fluid diameter changes, and hence, the value of slip velocity also changes.

Therefore, for different superficial velocities, either one has to take different dispersed fluid sizes, if drag law is constant; or select different drag laws, while keeping the dispersed fluid size constant.　The first

Drag models CFD (present work)

CD＝0.44 ×Schiller and Naumann43) △Grace et al.44) ◇Ishii and Zuber30) ■

Fig. 2　Comparison of Various Drag Models at Axial Position of 1.25 m

Table 3　Drag Laws Considered

Author Model

24 0 687 C = (1 + 0 15 Re ) if Re = 1000Schiller and D

pp pRe

. ,.

Naumann43)

CD = 0 44 . , if Re p = 1000

4 gdp (rq − rp )Grace et al.44) CD = 3 vq

2 rq

2 0 5.Ishii and Zuber30) CD = E03

option is more realistic and represents the actual physi-cal picture.　Hence, it is always advisable to change the dispersed fluid size with change in superficial veloc-ity.　This elucidates the issue of proper choice of com-bination of drag law and dispersed fluid size as far as CFD simulation of vertical pipe hydrodynamics is con-cerned.　Now, the drag force alone will not be suffi-cient to predict the local hydrodynamics correctly.　


48

Proper description of other interface forces such as lift force and dispersion force, are also important.

Therefore, to understand the effect of different inter-phase forces, a series of simulations have been carried out.　For this purpose, the experimental data reported by Farrar and Bruun21) have been considered. 4. 1.　Effect of Drag Law

Simulations with all the drag laws given in Table 3 have been carried out.　The dispersed fluid sizes for all simulations have been chosen in such away to satisfy the average dispersed phase holdup.　The various lift coefficients for the corresponding dispersed fluid size has been taken into account.　Standard parameter set-tings are given in Table 4.

It can be seen from Fig. 3 that though all the drag relations have been able to predict the average dispersed phase holdup and the centerline velocity quite accurately, the drag law reported by Ishii and Zuber30) is closer to the experimental values. 4. 2.　Effect of Lift Force

To study the effect of lift force, various values of CL

were chosen for the case given in Table 4.　Simulations

Drag models CFD (present work) Exp. F20

CD＝0.44 ×Schiller and Naumann42) △

●Grace et al.43) ◇Ishii and Zuber30) ■

Fig. 3　Effect of Various Drag Models on the Dispersed Phase Hold-up at Axial Position of 1.25 m for the Data Set of Farrar and Bruun21)

have been carried out with the value of CL＝0 and the corresponding positive and negative values (CL＝0.005, 0.001, 0.01, 0.1 and CL＝－0.01, －0.005, －0.003,－0.002), while other parameter values were same as that reported in Table 4.　The results for holdup pro-file have been given in Figs. 4a, 4b, 4c and 4d.

While positive value of CL tends to predict increasing wall peaks, negative constants shift the dispersed phase holdup profiles towards an increased coring tendency.　Referring to Fig. 4a, it can be seen that apart from the asymmetric nature of the holdup profiles when positive and negative constants of the same magnitude are used, the use of lift coefficients (i.e.＋0.01 and－0.01) results in very low and high dispersed phase holdups at the wall.　The flat holdup distribution for CL＝0 corre-sponds to the case when only drag forces are accounted for.

However, lift force with a negative coefficient has to be used for the case of the dispersed fluid in order to capture the trends observed in experiments.　It can be seen from Fig. 4b that the lift force can best be predicted using a negative coefficient (CL＝－0.003).　Figure 4c shows the effect of positive lift coefficient on low dispersed phase holdup for the data set of Al-Deen and Bruun22).　As expected, positive value of lift co-efficients are closer to experimental data than negative values shown at Fig. 4d.　As illustrated in Fig. 4c the closest value for lower holdup is CL＝0.01. 4. 3.　Effect of Turbulent Dispersion

Simulations were carried out with four values of CTD

based on Lopez de Bertodano’s constant (0, 0.2, 0.5 and 1), and all the other parameters were kept the same as reported in Table 4.

Figure 5 shows that at low phase holdup although, CTD value of 0.9 based on Favre average drag force was found to give a better agreement, but we feel the choice of the value CTD is intuitive, depending on the system under consideration, which can predict the holdup and axial liquid velocity profile closer to the experimental value.

It can be seen from Figs. 6a and 6b that at high phase fraction, the effect of CTD differs with the sign of lift coefficient; however, it gives a completely different pre-diction near the wall when positive or negative lift co-efficient is used.　As a result, the effect of dispersion at higher holdups is not that significant, but at lower holdups, increase in value of CTD makes the holdup pro-

Table 4　Standard Parameters Setting for Simulation (Farrar and Bruun21) and Al-Deen22))

Farrar and Bruun (1996) Al-Deen (1997)

Velocity [exp.] [m/s] 0.6169 0.5727 Dispersed fluid diameter [mm] 5 3 Lift force coefficient －0.003 0.01 Turbulent dispersion force coefficient 0 0.9 Added mass force coefficient — —


49

CL CFD (present work) Exp. F20CL CFD (present work) Exp. F20

◇□△×

－0.01－0.002－0.003－0.005

0 0.001 0.005 0.01

◆■×△

●●

Fig. 4a　Effect of Positive Lift Coefficient on Dispersed Phase Hold-Fig. 4b　Effect of Negative Lift Coefficient on Dispersed Phase

Holdup at Axial Position of 1.25 m for the Data Set of up at Axial Position of 1.25 m for the Data Set of Farrar and Farrar and Bruun21)

Bruun21)

CL CFD (present work) Exp. A05

CL CFD (present work) Exp. A05 －0.1　－0.01

●

□▲◇＋

□△×◆

0 0.001 0.005 0.01

－0.002● －0.003

－0.005 －

Fig. 4d　Effect of Negative Lift Coefficient on Low Dispersed Phase Fig. 4c　Effect of Positive Lift Coefficient on Low Dispersed Phase Holdup for the Data Set of Al-Deen and Bruun22) (Favre 0.9 Holdup for the Data Set of Al-Deen and Bruun22) (Favre 0.9 for CTD, lift constant varies and axial testing position is for CTD, lift constant varies and axial testing position is 1.25 m) 1.25 m)


50

CL Favre CFD (present work) Exp. A05

0.01 0.7 ◇0.04 0.9 ■ ●0.09 0.9 ×

Fig. 5　Simulated Phase Distribution Profiles for the Data Set of Al-Deen and Bruun22) (Favre for CTD, lift constant varies and axial testing position is 1.25 m)

file comparatively flatter. 4. 4.　Effect of Added or Virtual Mass Force

In fundamental nature, the virtual mass force is the force required to accelerate mass of the continuous phase in the immediate vicinity of the dispersed phase.　It has the effect of dampening the natural tendency of the particle to accelerate in any direction.　This force can be important near dispersed phase inlets and outlets and plays a particularly significant role when the density of the dispersed phase is much smaller than that of the continuous phase.　It therefore is important in the case of bubbles and has no important effect on kerosene-water system (there are not much difference between densities of these two liquids).　To observe the conse-quence of virtual mass, three simulations were carried out with CVM＝0, 0.5 and 0.7.　The results are given in Fig. 7.　True to the previous prediction, it can be seen that, the addition of added mass force has no significant effect on results. 4. 5.　Effect of Various Droplet Diameter

Most of the studies are focused on accounting for uniform size of droplets.　As exact size distribution and even exact mean diameter for dispersed phase were not prescribed by any of investigators in their experi-ments, the equivalent drop diameter had to be guessed.　Since previously mentioned in Eq. (20), some important parameters such as slip velocity and Reynolds number differ with diameter of drops.　As shown in Figs. 8 and 9, the dispersed phase and continuous velocity profile vary with equivalent diameter at different radial posi-

CL Lopez CFD (present work) Exp. F20

－0.003 0.1 × ●0.003 0.1 ◇

Fig. 6a　Simulated Phase Distribution Profiles for the Data Set of Farrar and Bruun21) (Lopez 0.1, lift constant varies and axial position of 1.25 m)

CL Lopez CFD (present work) Exp. F20

－0.003 0.5 ×0 0.5 □ ●0.003 0.5 ▲

Fig. 6b　Simulated Phase Distribution Profiles for the Data Set of Farrar and Bruun21) (Lopez 0.5, lift constant varies and axial position of 1.25 m)

tions.　Then error of simulations for small drops can be neglected, but for accurate simulations, drop size distribution should be predicted and validated.　Then


51

CVM CFD (present work) Exp. F20

0 □0.5 △ ●0.7 ＋

Fig. 7　Effect of Virtual Mass Coefficient on Dispersed Phase Hold-up at Axial Position of 1.25 m for the Data Set of Farrar and Bruun21) (Ishii and Zuber30) expression for drag coefficient and－0.003 for lift coefficient)

Radial positions [m] CFD (present work)

0 ◆0.009 □0.015 ●0.021 △0.027 ×0.03 ＋0.033 ○

Fig. 8　Effect of Varying of Drop Diameter on the Holdup at Different Radial Positions and Axial Position of 1.25 m (Ishii and Zuber30) expression for drag coefficient and －0.003 for lift coefficient)

simulation can be done accounting for size distribution of droplets.　In these cases the use of methods such as population balance method is recommended.

Radial positions [m] CFD (present work)

0 ◇0.027 □0.03 △0.033 ×

Fig. 9　Effect of Varying of Drop Diameter on the Continuous Phase Velocity at Different Radial Positions and Axial Position of 1.25 m (Ishii and Zuber30) expression for drag coefficient and－0.003 for lift coefficient)

4. 6.　 Effect of Turbulence Model The choice of a turbulence model depends on consid-

erations such as the physics involved in the flow, the level of accuracy desired and the available computa-tional resources.　In the context of turbulent flows, the two-equation standard k-ε turbulence model is the most extensively studied and is, therefore, used as a baseline approach in RANS-based models.　With specific refer-ence to immiscible liquid dispersions, irrespective of the domain geometry, it is seen that the two-equation standard k-ε turbulence model is the most successful (reasonable accuracy, numerically robust and computa-tionally economical) and, therefore, extensively used.

In this paper turbulence was modeled using standard k-ε turbulence model.　An alternative approach (Reynolds Stress Model or RSM) was also tested.　Figures 10 and 11 show the comparison for radial pro-file of dispersed phase across the pipe for two different turbulence models at the experimental conditions of Farrar and Bruun21) and Al-Deen and Bruun22).　The RSM did not show better predictive performance than k-ε in predicting the average dispersed phase holdup.　It is clear that the best turbulent model to apply at this case for all simulations is k-ε.

5.　Conclusions

(1) For small equivalent drop diameter (≤3 mm), all available drag expressions for entities predict essentially similar holdups.　On the other hand, significantly lower dispersed phase holdups were predicted when the drag expression for a rigid sphere was used in conjunction


52

Turbulent models CFD (present work) Exp. F20

k-ε ▲ ●RSM □

Fig. 10　Effect of Various Turbulent Models on the Dispersed Phase Holdup at Axial Position of 1.25 m for the Data Set of Farrar and Bruun21) (Ishii and Zuber30) expression for drag coefficient and－0.003 for lift coefficient)

Turbulent models CFD (present work) Exp. F20

k-ε ▲ ●RSM □

Fig. 11　Effect of Various Turbulent Models on the Dispersed Phase Holdup at Axial Position of 1.25 m for the Data Set of Al-Deen and Bruun22) (Ishii and Zuber30) expression for drag coefficient and－0.003 for lift coefficient)

with equivalent diameters larger than 5 mm.　These are encountered at higher dispersed phase holdups where the increased coalescence tendency shifts the drop size distribution towards larger drops. (2) The various drag expressions for drops predict

essentially similar holdups.　The drag expression pro-posed by Ishii and Zuber30) was selected as a represen-tative of the all drop drag expressions. (3) Lift forces were found to play a significant role at both low and high dispersed phase holdups and their use is necessary to predict the experimentally observed wall peaking and coring profile.　Lift force with a neg-ative coefficient has to be used for the case of the bub-bles in order to capture the trends observed in experi-ments.　Then the lift forces can best be predicted using a negative coefficient －0.003 for high phase fractions and a positive coefficient 0.01 for low phase fractions. (4) Dispersion forces act to spread out the dispersed phase in the solution domain thus counteracting lift forces.　Turbulent dispersion was found to be signifi-cant at low dispersed phase holdups but is practically negligible at dispersed phase ratio larger than 25%.　Similar conclusions apply for the dispersion model pro-posed by Lopez de Bertodano32). (5) At lower holdup range (≤5%) positive lift coefficient with a turbulent dispersion constant CTD of 0.9 base on Favre41) yields a better match with the data whereas in the case of higher holdups a constant lift coefficient of－0.003 without the turbulent dispersion produces a good agreement with the experimental data. (6) No significant contribution of virtual mass force on the simulation of dispersed phase holdup in vertical pipe flow was seen. (7) The performance of two turbulence models (standard k-ε and RSM) were compared with the experimental data of Farrar and Bruun21) and Al-Deen and Bruun22).　The RSM did not show better predictive performance than k-ε in predicting the average dispersed phase holdup.

Acknowledgment We would like to acknowledge the fellowship help

given by Srinath Madhavan.

Nomenclatures

C : momentum transfer coefficient [—] CD : drag force coefficient [—] CL : lift force coefficient [—] Cm : constant in k-ε model [—] CS : Smagorinsky constant [—] CTD : turbulent dispersion coefficient [—] CVM : virtual mass force coefficient [—] d : drop diameter [m] E0 : Eotvos number [—] F : interface force [N] g : gravitational constant [m/s2] G : generation term [kg/m・s2] k

・: turbulent kinetic energy per unit mass [m2/s2]

m・ : mass flow rate [kg/s] P : pressure [N/m2] P’ : exact production term [kg/m・s3] Re : Reynolds number [—] S : source term [—] t : time [s]


53

v : velocity vector [m/s]vs : slip velocity [m/s]<Greeks>α : fractional phase holdup [—]μ : shear viscosity [Pa・s]ρ : density [kg/m3]R´ ´i : Reynolds stresses [N/m2] σ : Prandtl number for turbulent kinetic energy [—] τ : shear stress [Pa] λ : bulk viscosity [Pa・s] φ : turbulent kinetic energy [m2/s3] Φ : pressure-strain correlation [kg/m・s3] <Subscripts>

v vj

p : dispersed phase q : continuous phase t : turbulent<Superscript>— : mean component’ : fluctuating component→ : vector quantity

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54

要　　　旨

垂直管内流における油分散系の数値シミュレーション

Mahdi Parvini, Bahram Dabir, and Seyed Abolfazl Mohtashami

Dept. of Chemical Engineering, Amirkabir University of Technology, Hafez St., P.O. Box 15875/4413 Tehran, IRAN

非混合液体分散系を取り扱ったシステムは石油化学業界で広く用いられている。そして，石油化学の分野では界面熱・物質移動率は系全体の現象に強く影響を与えることが知られており，相間界面接触面積に依存している。本研究は垂直管における非混合液体分散系の数値流体解析（CFD）を行ったものである。二流体分散系に関しては二流体モデル（オイラー-オイラーモデル）が多相モデリング用に広く使用されており，特に適切な計算量で広範囲なホールドアップの詳細予測が必要な場合に使用される。連続相に水，分散相にケロセンを使用した CFD

シミュレーション結果，Farrarと Bruunの実験結果，ならびにAl-DeenとBruunの実験結果との比較を行った。これらのシミュ

レーションは乱流液体分散系において作用する抗力，揚力，乱流分散力，付加質量力等のさまざまな力の影響を確認するためにも行われた。最適なパラメーターの組合せを選定するため，これらの流体に作用する力を示すいくつかのモデルを使用したシミュレーションを行った。さらに，二つの異なる乱流モデルを使用したシミュレーションも行った。本研究の成果として，揚力が乱流分散力や付加質量力よりも重要であることが分かった。また，実験データと合致するシミュレーション結果をもとに，液液分散流における相間界面モデル化の指針を示すことができた。


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